Question

$\overline{a},\overline{b},\overline{c}$ are three non-zero vectors such that any two of them are non-collinear. If $\overline{a}+\overline{b}$ is collinear with $\overline{c}$ and $\overline{b}+\overline{c}$ is collinear with $\overline{a}$, then what is their sum?

• A. $-1$
• B. $0$
• C. $1$
• D. $2$

Answer 1

The correct option is B $0$

We have

$\overline{a}+\overline{b}=t\overline{c}$ ----$\left(1\right)$
$\overline{b}+\overline{c}=s\overline{a}$ ----$\left(2\right)$
From $\left(1\right)$ and $\left(2\right)$
$\overline{a}+\overline{b}=t(s\overline{a}-\overline{b})$
Since no two of them are collinear, comparing coeffficients gives
$st=1$ and $t=-1$
$\Rightarrow s=-1$ and $t=-1$
From $\left(1\right)$
$\therefore \overline{a}+\overline{b}+\overline{c}=0$
Hence, option $B$.